Some New Aspects in Hypercomplex Analysis

The aim of this overview article is to inform the reader on some new aspects in the use of hypercomplex methods. We have selected six topics, which are interesting for applications to physics, hydrodynamics, texture analysis, and mathematics itself. In the first part we introduce hyperquaternions and show its usefulness for the algebraical representation of the different kind of physics (P. Girard). The second part is devoted to the analysis on the 3-sphere and its practical applications to problems in texture analysis (H. Schaeben, R. Hielscher). In the third section we describe the fluid flow through porous media with the help of a quaternion operator calculus. For this reason the basics of the calculus ((D + α)-holomorphic functions, Bergman–Hodge decomposition of the quaternionic Hilbert space and its projections) has to be introduced (K. Guerlebeck, W. Sproessig). With the help of the Takenaka–Malmquist system a generalized Fourier expansion with the so-called mono-components is shortly described (T. Qian). Furthermore, harmonic conjugates in weighted Bergman spaces of quaternion valued functions are presented (K. Avetisyan). The harmonic conjugates of the Poisson kernel are given explicitly in ℝ 3 $$\mathbb {R}^3$$ and ℝ 4 $$\mathbb {R}^4$$ , which are higher dimensional generalizations of the famous Schwarz formula of the classic function theory.